Operations & Word Problems

Depth · one concept from first principles · step 2 of 4

Swapping mixtures shrinks the concentration gap

Two containers of salt water, 500 g each, trade 100 g one way and then 100 g back. Compare the final gap in concentration with the original gap.
RatiosExpressions
Your answer
How to solve
Strategy Convert to Algebra — Call the original concentrations x% and y%. Track grams of salt, not percents: salt adds up when solutions are mixed, percents do not. Write each final concentration in x and y, subtract, and compare with x - y.
1STEP 1

Pour A into B

A keeps 400 g at x%. B now has 600 g holding 5y grams of salt from itself and x grams from A.

B = (5y + x)/600 × 100 = (x + 5y)/6 (%)
2STEP 2

Pour B back into A

Taking 100 g from B keeps B's concentration. That 100 g carries (x+5y)/6 g of salt into A, which already has 4x g in 400 g.

A = (4x + (x+5y)/6)/500 × 100 = (24x + x + 5y)/30 = (5x + y)/6 (%)
3STEP 3

Subtract

The final gap is A minus B.

(5x + y)/6 - (x + 5y)/6 = (4x - 4y)/6 = (2/3)(x - y)
4STEP 4

Form the ratio

Divide by the original gap x - y, which is not zero because the concentrations differ.

(2/3)(x-y) : (x-y) = 2 : 3
Answer
2:3
Try A at 12% and B at 6%: B becomes (30 + 12)/600 = 7%, and A becomes (48 + 7)/500 = 11%. The gap goes from 6 to 4, and 4 : 6 = 2 : 3.
Takeaway

Track grams of salt, not percents, and the unknown concentrations cancel out.

  • Pour A into B
  • Pour B back into A
  • Subtract
  • Form the ratio
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